A weak solvability to the steady Navier–Stokes equations for compressible barotropic fluid with generalized impermeability boundary conditions

Abstract

We prove the existence of a steady solution to the Navier–Stokes equations for barotropic compressible fluid in a bounded simply connected domain with the prescribed generalized impermeability conditions u · n = 0, curl u · n = 0 and curl 2 u · n = 0 on the boundary, we assume that the state law for the pressure has the form P(ρ) = ργ for . We prove several auxiliary lemmas, e.g. on solution of the Stokes problem with the generalized impermeability boundary conditions in W 2, p (Ω) or on the extension of the equation of continuity satisfied in the sense of distributions from 𝒟′(Ω) to 𝒟′(ℝ3) for velocity with the normal component on the boundary of Ω equal to zero.